Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps

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June 2016, 36(6): 3375-3416. doi: 10.3934/dcds.2016.36.3375

Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps

Weiyuan Qiu ^1, , Fei Yang ^2, and Yongcheng Yin ^3,

1.	School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2.	Department of Mathematics, Nanjing University, Nanjing 210093, China
3.	Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received April 2015 Revised October 2015 Published December 2015

Abstract
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We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable parameters. Combining a result obtained before, we give a complete classification of the Cantor circles Julia sets in the sense of quasisymmetric equivalence. Moreover, we study the regularity of the components of the Cantor circles Julia sets and establish a sufficient and necessary condition when a component of a Cantor circles Julia set is a quasicircle.

Keywords: quasisymmetrically equivalent., Cantor circles, Julia sets.

Mathematics Subject Classification: Primary: 37F45; Secondary: 37F20, 37F1.

Citation: Weiyuan Qiu, Fei Yang, Yongcheng Yin. Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3375-3416. doi: 10.3934/dcds.2016.36.3375

References:

[1]	A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4422-6.
[2]	M. Bonk, Uniformization of Sierpiński carpets in the plane, Invent. Math., 186 (2011), 559-665. doi: 10.1007/s00222-011-0325-8.
[3]	M. Bonk, M. Lyubich and S. Merenkov, Quasisymmetries of Sierpiński carpet Julia sets, preprint,, , ().
[4]	M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, (French) [Hyperbolic buildings, conformal dimension and Mostow rigidity], Geom. Funct. Anal., 7 (1997), 245-268. doi: 10.1007/PL00001619.
[5]	M. Bourdon and H. Pajot, Quasi-conformal geometry and hyperbolic geometry, in Rigidity in Dynamics and Geometry, Springer, Berlin, 2002, 1-17.
[6]	G. Cui, Dynamics of rational maps, topology, deformation and bifurcation, Preprint, May, 2002 (early version: Geometrically finite rational maps with given combinatorics, 1997).
[7]	R. L. Devaney, D. Look and D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J., 54 (2005), 1621-1634. doi: 10.1512/iumj.2005.54.2615.
[8]	A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Éc Norm. Sup., 18 (1985), 287-343.
[9]	M. Gromov, Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75-263. doi: 10.1007/978-1-4613-9586-7_3.
[10]	P. Haïssinsky, Géométrie quasiconforme, analyse au bord des espaces métriques hyperboliques et rigidités, Astérisque, 326 (2009), 321-362.
[11]	P. Haïssinsky and K. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., 28 (2012), 1025-1034. doi: 10.4171/RMI/701.
[12]	J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0131-8.
[13]	M. Kapovich and B. Kleiner, Hyperbolic groups with low-dimensional boundary, Ann. Sci. Éc Norm. Sup., 33 (2000), 647-669. doi: 10.1016/S0012-9593(00)01049-1.
[14]	B. Kleiner, The asymptotic geometry of negatively curved spaces: Uniformization, geometrization and rigidity, in International Congress of Mathematicians, II, Eur. Math. Soc., Zürich, 2006, 743-768.
[15]	O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin, Heidelberg, New York, 1973.
[16]	C. McMullen, Automorphisms of rational maps, in Holomorphic Functions and Moduli I, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988, 31-60. doi: 10.1007/978-1-4613-9602-4_3.
[17]	J. Milnor, Dynamics in One Complex Variable: Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006.
[18]	K. Pilgrim and L. Tan, Rational maps with disconnected Julia sets, Astérisque, 261 (2000), 349-383.
[19]	W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577. doi: 10.1016/j.aim.2011.12.026.
[20]	W. Qiu, F. Yang and Y. Yin, Rational maps whose Julia sets are Cantor circles, Ergod. Th. & Dynam. Sys., 35 (2015), 499-529. doi: 10.1017/etds.2013.53.
[21]	N. Steinmetz, On the dynamics of the McMullen family $R(z)=z^m+\lambda/z^l$, Conform. Geom. Dyn., 10 (2006), 159-183. doi: 10.1090/S1088-4173-06-00149-4.
[22]	L. Tan and Y. Yin, Local connectivity of the Julia sets for geometrically finite rational maps, Sci. China Ser. A, 39 (1996), 39-47.

show all references

References:

[1]	A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4422-6.
[2]	M. Bonk, Uniformization of Sierpiński carpets in the plane, Invent. Math., 186 (2011), 559-665. doi: 10.1007/s00222-011-0325-8.
[3]	M. Bonk, M. Lyubich and S. Merenkov, Quasisymmetries of Sierpiński carpet Julia sets, preprint,, , ().
[4]	M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, (French) [Hyperbolic buildings, conformal dimension and Mostow rigidity], Geom. Funct. Anal., 7 (1997), 245-268. doi: 10.1007/PL00001619.
[5]	M. Bourdon and H. Pajot, Quasi-conformal geometry and hyperbolic geometry, in Rigidity in Dynamics and Geometry, Springer, Berlin, 2002, 1-17.
[6]	G. Cui, Dynamics of rational maps, topology, deformation and bifurcation, Preprint, May, 2002 (early version: Geometrically finite rational maps with given combinatorics, 1997).
[7]	R. L. Devaney, D. Look and D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J., 54 (2005), 1621-1634. doi: 10.1512/iumj.2005.54.2615.
[8]	A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Éc Norm. Sup., 18 (1985), 287-343.
[9]	M. Gromov, Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75-263. doi: 10.1007/978-1-4613-9586-7_3.
[10]	P. Haïssinsky, Géométrie quasiconforme, analyse au bord des espaces métriques hyperboliques et rigidités, Astérisque, 326 (2009), 321-362.
[11]	P. Haïssinsky and K. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., 28 (2012), 1025-1034. doi: 10.4171/RMI/701.
[12]	J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4613-0131-8.
[13]	M. Kapovich and B. Kleiner, Hyperbolic groups with low-dimensional boundary, Ann. Sci. Éc Norm. Sup., 33 (2000), 647-669. doi: 10.1016/S0012-9593(00)01049-1.
[14]	B. Kleiner, The asymptotic geometry of negatively curved spaces: Uniformization, geometrization and rigidity, in International Congress of Mathematicians, II, Eur. Math. Soc., Zürich, 2006, 743-768.
[15]	O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin, Heidelberg, New York, 1973.
[16]	C. McMullen, Automorphisms of rational maps, in Holomorphic Functions and Moduli I, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988, 31-60. doi: 10.1007/978-1-4613-9602-4_3.
[17]	J. Milnor, Dynamics in One Complex Variable: Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006.
[18]	K. Pilgrim and L. Tan, Rational maps with disconnected Julia sets, Astérisque, 261 (2000), 349-383.
[19]	W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577. doi: 10.1016/j.aim.2011.12.026.
[20]	W. Qiu, F. Yang and Y. Yin, Rational maps whose Julia sets are Cantor circles, Ergod. Th. & Dynam. Sys., 35 (2015), 499-529. doi: 10.1017/etds.2013.53.
[21]	N. Steinmetz, On the dynamics of the McMullen family $R(z)=z^m+\lambda/z^l$, Conform. Geom. Dyn., 10 (2006), 159-183. doi: 10.1090/S1088-4173-06-00149-4.
[22]	L. Tan and Y. Yin, Local connectivity of the Julia sets for geometrically finite rational maps, Sci. China Ser. A, 39 (1996), 39-47.

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